I am Emily Josephs

• interested in: evolutionary genetics, plants, triathlons, cat

I am Emily Josephs

• interested in: evolutionary genetics, plants, triathlons, cat

1. Data into spreadsheet

2. Wrangle data into R

3. Visualize data

4. Build a model

   • how are data distributed?
   • relationship between predictor and response?

5. Use algorithm to parameterize model

6. Interpret results

1. Data into spreadsheet

2. Wrangle data into R

3. Visualize data

4. Build a model

   • how are data distributed?
   • relationship between predictor and response?

5. Use algorithm to parameterize model

6. Interpret results

Office hours are Mondays 2-4pm in PLB 266 or on zoom.

Zoom Room: https://msu.zoom.us/my/emjos (Password: plantsrule)

Simulations?

One of the most powerful tools that we’ll have in our statistics learning toolkit are simulations.

Simulations let you generate data that you know should look a certain way, so you can test your intuitions.

Simulations also let you do the same thing over and over and over.

If we randomly pick a student, how likely are we to select student #19?

If we randomly pick 10 students, how likely are we to select student #19?

If we randomly pick 2 students, how likely are we to select students #19 and #20?

If we randomly pick a student, how likely are we to select student #19?

mySamples <- replicate(10000, sample(thisClass, size=1))
sum(mySamples==19)/10000

## [1] 0.0304

The sample space for this example is that ( A), The ball can fall through the orange bin, ( B) the ball can fall through the green bin, and ( C) the ball can fall through the blue bin

Work with your groups to calculate the probability of A or B happening.

Note that this is called a complex event, since it is the combination of multiple, mutually-exclusive outcomes.

What if we want to simulate many samples to get a sampling distribution of how likely we are to get A or B?

nReps <- 100

sampleFunction <- function(myN){
                  mySample = sample(x=c("A","B","C"), 
                  size = myN,
                  replace=TRUE,
                  prob = c(pA, pB, pC))
                  
                  propA = sum(mySample=="A")/myN
                  propB = sum(mySample=="B")/myN
                  propC = sum(mySample=="C")/myN
                  
                  return(propA + propB)
                  }

mySamples <- replicate(nReps, sampleFunction(500))

hist(mySamples, main="", xlab = "proportion A or B")

How can we edit this sample function to return the probability of falling through both A and B if \(P(A) = 0.5\) and \(P(B) = 0.5\)?

sampleFunction <- function(myN){
                  mySample = sample(x=c("A","B","C"), 
                  size = myN,
                  replace=TRUE,
                  prob = c(pA, pB, pC))
                  
                  propA = sum(mySample=="A")/myN
                  propB = sum(mySample=="B")/myN
                  propC = sum(mySample=="C")/myN
                  
                  return(propA + propB)
                  }

pA = pB = 0.5

sampleFunction2 <- function(myN){
                  mySampleA = sample(x=c(1,0), 
                  size = myN, replace=TRUE,
                  prob = c(pA, 1-pA))
                  
                  mySampleB = sample(x=c(1,0), 
                  size = myN, replace=TRUE,
                  prob = c(pB, 1-pB))
                  
                  myCombined = mySampleA + mySampleB
                  
                  probBoth = sum(myCombined==2)/myN
                  
                  return(probBoth)
                  }

mySamples <- replicate(100,sampleFunction2(100))

hist(mySamples, main="", xlab = "proportion A and B")

Modify our sampling function to simulate the following scenario: \(P(A) = 1/3\), \(P(B|A) = 1/2\), \(p(B|not A) = 3/4\)

How often do we find A and B?

Note that this is called a shared event, made up of multiple non-exclusive outcomes.

pA = 1/2
pB_given_A    = 4/5
pB_given_notA = 1/5

#write a function to do one trial (one ball)

#write a function to get a sample by running that trial 500 times

#write a function to generate 100 samples

#write a function to do one trial (one ball)
oneTrial <- function(){
  mySampleA = sample(x=c(1,0), 
                  size = 1, replace=TRUE,
                  prob = c(pA, 1-pA))
                  
                  if (mySampleA == 1){pB = pB_given_A}
                  else{pB = pB_given_notA}
                  
                  mySampleB = sample(x=c(1,0), 
                  size = 1, replace=TRUE,
                  prob = c(pB, 1-pB))
                  
                  myCombined = mySampleA + mySampleB
                  isBoth = sum(myCombined==2)
                  return(isBoth)
}

#write a function to get a sample by running that trial 500 times

sampleFunction3 <- function(myN){sum(replicate(myN, oneTrial()))/myN}

#write a function to generate 100 samples
mySamples <- replicate(100,sampleFunction3(100))

hist(mySamples, main="", xlab = "Prob of A and B")

##Intro to Probability! {.build}

Learning goals:

Learn what probability is and the law of total probability.

Write basic simulations in R to build intuition about: -simple mutually-exclusive events -complex mutually-exclusive events -shared non-exclusive events -conditional probabilities